Step 1: Understanding the Question:
We need to find the effective inductance and capacitance of the circuit to calculate the resonant frequency. Step 2: Key Formula or Approach:
1. Resonant frequency \( f = \frac{1}{2\pi \sqrt{L_{\text{eff}} C_{\text{eff}}}} \).
2. Inductors in parallel: \( L_{\text{eff}} = \frac{L_1 L_2}{L_1 + L_2} \).
3. Capacitors in series: \( C_{\text{eff}} = \frac{C_1 C_2}{C_1 + C_2} \). Step 3: Detailed Explanation:
From the diagram, two inductors \( L \) are in parallel.
\[ L_{\text{eff}} = \frac{L \times L}{L + L} = \frac{L}{2} \]
Two capacitors \( C \) are in series.
\[ C_{\text{eff}} = \frac{C \times C}{C + C} = \frac{C}{2} \]
Substitute into the resonant frequency formula:
\[ f = \frac{1}{2\pi \sqrt{\left(\frac{L}{2}\right) \left(\frac{C}{2}\right)}} \]
\[ f = \frac{1}{2\pi \sqrt{\frac{LC}{4}}} = \frac{1}{2\pi \left(\frac{\sqrt{LC}}{2}\right)} \]
\[ f = \frac{1}{\pi \sqrt{LC}} \] Step 4: Final Answer:
The resonant frequency is \( \frac{1}{\pi \sqrt{LC}} \).
